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chebyshev.h
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chebyshev.h
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//==================================================================
/**
* chebyshev.h -- C++ functions to evaluate Chebyshev polynomials
*
* Copyright (C) 2019 by James A. Chappell (rlrrlrll@gmail.com)
*
* Permission is hereby granted, free of charge, to any person
* obtaining a copy of this software and associated documentation
* files (the "Software"), to deal in the Software without
* restriction, including without limitation the rights to use,
* copy, modify, merge, publish, distribute, sublicense, and/or
* sell copies of the Software, and to permit persons to whom the
* Software is furnished to do so, subject to the following
* condition:
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
* OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
* HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
* WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
* FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
* OTHER DEALINGS IN THE SOFTWARE.
*/
//=================================================================
/*
* chebyshev.h: Version 0.03
* Created by James A. Chappell <rlrrlrll@gmail.com>
* http://www.storage-b.com/math-numerical-analysis/27
* Created 28 July 2007
*
* History:
* 28-jul-2007 created
* 14-nov-2014 templates
* 15-nov-2019 deduced types
* 18-nov-2019 add second kind
*/
//==============
#ifndef __CHEBYSHEV_H__
#define __CHEBYSHEV_H__
namespace Storage_B
{
/*
* Functions calculate Chebyshev Polynomials Tn(x) and Un(x)
*/
namespace Chebyshev
{
// Chebyshev Polynomials of the first kind: Tn(x)
//
// n = 0
template <class T> inline auto T0(const T& x)
{
return static_cast<T>(1);
}
// n = 1
template <class T> inline auto T1(const T& x)
{
return x;
}
// n = 2
template <class T> inline auto T2(const T& x)
{
return (static_cast<T>(2) * x*x) - static_cast<T>(1);
}
/*
* Tn(x)
*/
template <class T> inline auto Tn(unsigned int n, const T& x)
{
switch(n)
{
case 0:
return T0<T>(x);
case 1:
return T1<T>(x);
case 2:
return T2<T>(x);
default:
break;
}
/* We could simply do this:
return (static_cast<T>(2) * x * Tn(n - 1, x)) - Tn(n - 2, x);
but it could be slow for large n */
auto tnm1(T2<T>(x));
auto tnm2(T1<T>(x));
auto tn(tnm1);
for (auto l = 3u ; l <= n ; ++l)
{
tn = (static_cast<T>(2) * x * tnm1) - tnm2;
tnm2 = tnm1;
tnm1 = tn;
}
return tn;
}
// Chebyshev Polynomials of the second kind: Un(x)
//
// n = 0
template <class T> inline auto U0(const T& x)
{
return static_cast<T>(1);
}
// n = 1
template <class T> inline auto U1(const T& x)
{
return static_cast<T>(2) * x;
}
// n = 2
template <class T> inline auto U2(const T& x)
{
return (static_cast<T>(4) * x*x) - static_cast<T>(1);
}
/*
* Un(x)
*/
template <class T> inline auto Un(unsigned int n, const T& x)
{
switch(n)
{
case 0:
return U0<T>(x);
case 1:
return U1<T>(x);
case 2:
return U2<T>(x);
default:
break;
}
//return (static_cast<T>(2) * x * Un<T>(n - 1, x)) - Un<T>(n - 2, x);
auto unm1(U2<T>(x));
auto unm2(U1<T>(x));
auto un(unm1);
for (auto l = 3u ; l <= n ; ++l)
{
un = (static_cast<T>(2) * x * unm1) - unm2;
unm2 = unm1;
unm1 = un;
}
return un;
}
}
}
#endif